Abstract

We propose a general method called convolution-variation separation (CVS) to enable efficient optical imaging calculations without sacrificing accuracy when simulating images for a wide range of process variations. The CVS method is derived from first principles using a series expansion, which consists of a set of predetermined basis functions weighted by a set of predetermined expansion coefficients. The basis functions are independent of the process variations and thus may be computed and stored in advance, while the expansion coefficients depend only on the process variations. Optical image simulations for defocus and aberration variations with applications in robust inverse lithography technology and lens aberration metrology have demonstrated the main concept of the CVS method.

© 2013 Optical Society of America

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References

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2012 (1)

2011 (3)

2009 (1)

2008 (1)

2007 (1)

A. Poonawala and P. Milanfar, IEEE Trans. Image Process. 16, 774 (2007).
[CrossRef]

2006 (2)

T. Brunner, D. Corliss, S. Butt, T. Wiltshire, C. P. Ausschnitt, and M. Smith, J. Microlith. Microfab. Microsyst. 5, 043003 (2006).
[CrossRef]

P. Yu, D. Z. Pan, and C. A. Mack, Proc. SPIE 6156, 397 (2006).

1994 (1)

1982 (1)

1979 (1)

B. E. A. Saleh, Opt. Acta 26, 777 (1979).
[CrossRef]

1978 (1)

1953 (1)

H. H. Hopkins, Proc. R. Soc. A 217, 408 (1953).

Ausschnitt, C. P.

T. Brunner, D. Corliss, S. Butt, T. Wiltshire, C. P. Ausschnitt, and M. Smith, J. Microlith. Microfab. Microsyst. 5, 043003 (2006).
[CrossRef]

Brunner, T.

T. Brunner, D. Corliss, S. Butt, T. Wiltshire, C. P. Ausschnitt, and M. Smith, J. Microlith. Microfab. Microsyst. 5, 043003 (2006).
[CrossRef]

Butt, S.

T. Brunner, D. Corliss, S. Butt, T. Wiltshire, C. P. Ausschnitt, and M. Smith, J. Microlith. Microfab. Microsyst. 5, 043003 (2006).
[CrossRef]

Cobb, N. B.

N. B. Cobb, “Fast optical and process proximity correction algorithms for integrated circuit manufacturing,” Ph.D. dissertation (University of California at Berkeley, 1998).

Corliss, D.

T. Brunner, D. Corliss, S. Butt, T. Wiltshire, C. P. Ausschnitt, and M. Smith, J. Microlith. Microfab. Microsyst. 5, 043003 (2006).
[CrossRef]

Davis, B. J.

Hopkins, H. H.

H. H. Hopkins, Proc. R. Soc. A 217, 408 (1953).

Jia, N.

Kailath, T.

Kintner, E.

Lam, E. Y.

Liu, S. Y.

S. Y. Liu, S. Xu, X. F. Wu, and W. Liu, Opt. Express 20, 14272 (2012).
[CrossRef]

S. Y. Liu, W. Liu, and T. T. Zhou, J. Micro/Nanolith. MEMS MOEMS 10, 023007 (2011).
[CrossRef]

Liu, W.

S. Y. Liu, S. Xu, X. F. Wu, and W. Liu, Opt. Express 20, 14272 (2012).
[CrossRef]

S. Y. Liu, W. Liu, and T. T. Zhou, J. Micro/Nanolith. MEMS MOEMS 10, 023007 (2011).
[CrossRef]

Mack, C. A.

P. Yu, D. Z. Pan, and C. A. Mack, Proc. SPIE 6156, 397 (2006).

Milanfar, P.

A. Poonawala and P. Milanfar, IEEE Trans. Image Process. 16, 774 (2007).
[CrossRef]

Miller, W.

W. Miller, Symmetry and Separation of Variables (Cambridge University, 1984).

Neureuther, A. R.

Pan, D. Z.

P. Yu, D. Z. Pan, and C. A. Mack, Proc. SPIE 6156, 397 (2006).

Pati, Y. C.

Poonawala, A.

A. Poonawala and P. Milanfar, IEEE Trans. Image Process. 16, 774 (2007).
[CrossRef]

Saleh, B. E. A.

B. E. A. Saleh, Opt. Acta 26, 777 (1979).
[CrossRef]

Schoonover, R. W.

Shen, Y.

Smith, M.

T. Brunner, D. Corliss, S. Butt, T. Wiltshire, C. P. Ausschnitt, and M. Smith, J. Microlith. Microfab. Microsyst. 5, 043003 (2006).
[CrossRef]

Starikov, A.

Wei, H.

H. Wei, “Computational efficiency in photolithographic process simulation,” U.S. patent7,788,628 (August31, 2010).

Wiltshire, T.

T. Brunner, D. Corliss, S. Butt, T. Wiltshire, C. P. Ausschnitt, and M. Smith, J. Microlith. Microfab. Microsyst. 5, 043003 (2006).
[CrossRef]

Wolf, E.

Wong, A. K.

A. K. Wong, Optical Imaging in Projection Microlithography (SPIE, 2005).

Wong, N.

Wu, X. F.

Xu, S.

Yamazoe, K.

Yu, P.

P. Yu, D. Z. Pan, and C. A. Mack, Proc. SPIE 6156, 397 (2006).

Zhou, T. T.

S. Y. Liu, W. Liu, and T. T. Zhou, J. Micro/Nanolith. MEMS MOEMS 10, 023007 (2011).
[CrossRef]

Appl. Opt. (2)

IEEE Trans. Image Process. (1)

A. Poonawala and P. Milanfar, IEEE Trans. Image Process. 16, 774 (2007).
[CrossRef]

J. Micro/Nanolith. MEMS MOEMS (1)

S. Y. Liu, W. Liu, and T. T. Zhou, J. Micro/Nanolith. MEMS MOEMS 10, 023007 (2011).
[CrossRef]

J. Microlith. Microfab. Microsyst. (1)

T. Brunner, D. Corliss, S. Butt, T. Wiltshire, C. P. Ausschnitt, and M. Smith, J. Microlith. Microfab. Microsyst. 5, 043003 (2006).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (2)

Opt. Acta (1)

B. E. A. Saleh, Opt. Acta 26, 777 (1979).
[CrossRef]

Opt. Express (2)

Opt. Lett. (1)

Proc. R. Soc. A (1)

H. H. Hopkins, Proc. R. Soc. A 217, 408 (1953).

Proc. SPIE (1)

P. Yu, D. Z. Pan, and C. A. Mack, Proc. SPIE 6156, 397 (2006).

Other (4)

A. K. Wong, Optical Imaging in Projection Microlithography (SPIE, 2005).

W. Miller, Symmetry and Separation of Variables (Cambridge University, 1984).

N. B. Cobb, “Fast optical and process proximity correction algorithms for integrated circuit manufacturing,” Ph.D. dissertation (University of California at Berkeley, 1998).

H. Wei, “Computational efficiency in photolithographic process simulation,” U.S. patent7,788,628 (August31, 2010).

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Figures (4)

Fig. 1.
Fig. 1.

(a) Desired SRAM pattern, and the optimized mask patterns obtained after 100 iterations under (b) defocus variation with a Gaussian distribution, (c) defocus variation with a modified Lorentzian distribution, and (d) nominal condition.

Fig. 2.
Fig. 2.

(a) EPEs and (b) E-D trees of the optimized mask patterns obtained, respectively, under the nominal condition and defocus variations with the Gaussian and the modified Lorentzian distributions.

Fig. 3.
Fig. 3.

Binary mask pattern and image intensity results under the aberration shown in Fig. 4(a).

Fig. 4.
Fig. 4.

Example of aberration reconstruction. (a) Unknown aberration and (b) reconstructed aberration, and (c) wavefront error.

Equations (11)

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I ( x ; v ) = t ( x x 1 , x x 2 ; v ) o ( x 1 ) o * ( x 2 ) d x 1 d x 2 ,
t ( x 1 , x 2 ; v ) = s ( x 1 x 2 ) p ( x 1 ; v ) p * ( x 2 ; v )
p ( x ; v ) = m a m ( v ) p m ( x ) ,
t ( x 1 , x 2 ; v ) = m c m ( v ) t m ( x 1 , x 2 ) ,
t m ( x 1 , x 2 ) = n μ m n ϕ m n φ m n * ,
I ( x ; v ) = m c m ( v ) I m ( x ) ,
I m ( x ) = n μ m n [ ϕ m n o ( x ) ] [ φ m n o ( x ) ] * ,
T ( L ) = T ( 1 ) × L .
T CVS ( L ) = T 0 + δ T × L .
o ^ = arg min o { k = 1 L [ ζ ( h k ) ( sig [ I ( x ; h k ) ] o ˜ 2 2 ) ] + η R ( o ) } ,
I ( x ; Z ) = I 0 ( x ) + I 1 ( x ) = I 0 ( x ) + k = 1 K Z k I lin ( k ) ( x ) ,

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